Cambridge IGCSE Mathematics - Additional 0606 — 2010 May/June Paper 2 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Write your answers on the separate Answer Booklet/Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. * 1 4 0 6 9 2 4 4 7 6 * ADDITIONAL MATHEMATICS 0606/21 Paper 2 May/June 2010 2 hours Additional Materials: Answer Booklet/Paper Electronic calculator This document consists of 6 printed pages and 2 blank pages. DC (LEO/KN) 25699/1 © UCLES 2010 [Turn over www.theallpapers.com

2 0606/21/M/J/10 © UCLES 2010 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x b b ac a = − − 2 4 2 . Binomial Theorem (a + b)n = an + ( n 1)an–1 b + ( n 2)an–2 b2 + … + ( n r)an–r br + … + bn, where n is a positive integer and ( n r) = n! (n – r)!r! . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A. Formulae for ∆ABC a sin A = b sin B = c sin C . a2 = b2 + c2 – 2bc cos A. ∆ = 1 2 bc sin A. www.theallpapers.com

3 0606/21/M/J/10 © UCLES 2010 [Turn over 1 O y–– x2 x3 × ×(7, 1) (3, 9) The variables x and y are related so that, when y–– x2 is plotted against x3, a straight line graph passing through (3, 9) and (7, 1) is obtained. Express y in terms of x. [4] 2 In a singing competition there are 8 contestants. Each contestant sings in the first round of this competition. (i) In how many different orders could the contestants sing? [1] After the first round 5 contestants are chosen. (ii) In how many different ways can these 5 contestants be chosen? [2] These 5 contestants sing again and then First, Second and Third prizes are awarded to three of them. (iii) In how many different ways can the prizes be awarded? [2] 3 It is given that x – 1 is a factor of f(x), where f(x) = x3 – 6x2 + ax + b. (i) Express b in terms of a. [2] (ii) Show that the remainder when f(x) is divided by x – 3 is twice the remainder when f(x) is divided by x – 2. [4] www.theallpapers.com

4 0606/21/M/J/10 © UCLES 2010 4 (a) Given that sin x = p and cos x = 2p, where x is acute, find the exact value of p and the exact value of cosec x. [3] (b) Prove that (cot x + tan x) (cot x – tan x) = 1 ––––– sin2 x – 1 ––––– cos2 x . [3] 5 Given that a curve has equation y = x2 + 64 x , find the coordinates of the point on the curve where d2y ––– dx2 = 0. [7] 6 The line y = x + 4 intersects the curve 2x2 + 3xy – y2 + 1 = 0 at the points A and B. Find the length of the line AB. [7] 7 Solutions to this question by accurate drawing will not be accepted. O y x C(4, 10) D A(–1, 5) B(–2, 6) In the diagram the points A(–1, 5), B(–2, 6), C(4, 10) and D are the vertices of a quadrilateral in which AD is parallel to the x-axis. The perpendicular bisector of BC passes through D. Find the area of the quadrilateral ABCD. [8] www.theallpapers.com

5 0606/21/M/J/10 © UCLES 2010 [Turn over 8 (a) Given that A =  2 3 7 1 –5 4 and B =  2 1 8 6, calculate (i) 2A, [1] (ii) B2, [2] (iii) BA. [2] (b) (i) Given that C =  2 1 7 6, find C–1. [2] (ii) Given also that D =  4 3 –2 –1, find the matrix X such that XC = D. [2] 9 A particle starts from rest and moves in a straight line so that, t seconds after leaving a fixed point O, its velocity, v ms–1, is given by v = 4 sin 2t. (i) Find the distance travelled by the particle before it first comes to instantaneous rest. [5] (ii) Find the acceleration of the particle when t = 3. [3] 10 In this question,  1 0 is a unit vector due east and  0 1 is a unit vector due north. A lighthouse has position vector  27 48 km relative to an origin O. A boat moves in such a way that its position vector is given by  4 + 8t 12 + 6t km, where t is the time, in hours, after 1200. (i) Show that at 1400 the boat is 25 km from the lighthouse. [4] (ii) Find the length of time for which the boat is less than 25 km from the lighthouse. [4] www.theallpapers.com

6 0606/21/M/J/10 © UCLES 2010 11 Answer only one of the following two alternatives. EITHER B O A C D 0.8π 6 cm The diagram represents a company logo ABCDA, consisting of a sector OABCO of a circle, centre O and radius 6 cm, and a triangle AOD. Angle AOC = 0.8π radians and C is the mid-point of OD. Find (i) the perimeter of the logo, [7] (ii) the area of the logo. [5] OR y O P(1, 8) y = x3 – 6x2 + 8x + 5 Q x The diagram shows part of the curve y = x3 – 6x2 + 8x + 5. The tangent to the curve at the point P(1, 8) cuts the curve at the point Q. (i) Show that the x-coordinate of Q is 4. [6] (ii) Find the area of the shaded region. [6] www.theallpapers.com

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8 0606/21/M/J/10 © UCLES 2010 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. www.theallpapers.com

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2010 question paper for the guidance of teachers 0606 ADDITIONAL MATHEMATICS 0606/21 Paper 21, maximum raw mark 80 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes must be read in conjunction with the question papers and the report on the examination. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.theallpapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0606 21 © UCLES 2010 Mark Scheme Notes Marks are of the following three types: M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Accuracy mark for a correct result or statement independent of method marks. • When a part of a question has two or more "method" steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. • The symbol √ implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously "correct" answers or results obtained from incorrect working. • Note: B2 or A2 means that the candidate can earn 2 or 0. B2, 1, 0 means that the candidate can earn anything from 0 to 2. www.theallpapers.com

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0606 21 © UCLES 2010 The following abbreviations may be used in a mark scheme or used on the scripts: AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) CAO Correct Answer Only (emphasising that no "follow through" from a previous error is allowed) ISW Ignore Subsequent Working MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) Penalties MR -1 A penalty of MR -1 is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become "follow through √" marks. MR is not applied when the candidate misreads his own figures – this is regarded as an error in accuracy. OW -1,2 This is deducted from A or B marks when essential working is omitted. PA -1 This is deducted from A or B marks in the case of premature approximation. S -1 Occasionally used for persistent slackness – usually discussed at a meeting. EX -1 Applied to A or B marks when extra solutions are offered to a particular equation. Again, this is usually discussed at the meeting. www.theallpapers.com

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0606 21 © UCLES 2010 1 Gradient –2 B1 y intercept 15 B1 15 2 3 2 + − = x x y M1 2 5 15 2 x x y + − = A1√ [4] 2 (i) 40320 B1 (ii) ( )1 2 3 4 5 4 5 6 7 8 × × × × × × × × or !3 !5 !8 × M1 56 A1 (iii) uses 5, 4 and 3 only M1 60 A1 [5] 3 (i) f(1) = 0 M1 b = 5 – a only A1 (ii) Finds f(2) and f(3) M1 substitute b = 5 – a or a = 5 – b M1 2a – 22 or a – 11 (–2b – 12 or –b – 6) A1 both correct A1 [6] 4 (a) p2 + (2p)2 = 1 or cot x = 2 B1 5 1 = p or 5 cosec cot 1 cosec 2 2 = ⇒ + = x x x B1 5 cosec = x or 5 1 = p B1√ (b) cot2 x – tan2 x oe B1 cosec2 x – 1 – (sec2 x – 1) or other relevant use of Pythagoras M1 Correctly reaches conclusion x x 2 2 cos 1 sin 1 − A1 [6] 5 2 1 32 2 − + x x B1+B1 2 3 16 2 − − x B1+B1 Equate to 0 and solve M1 4 = x A1 144 = y A1 [7] www.theallpapers.com

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0606 21 © UCLES 2010 6 Eliminate x or y M1 4x2 + 4x – 15 = 0 or 4y2 – 28y + 33 = 0 A1 Factorise 3 term quadratic or use formula M1 2 3 = x and 2 5 − A1 2 11 = y and 2 3 A1√ 2 2 4 4 + M1 32 or 2 4 or 5.66 A1 [7] 7 Midpoint (1, 8) B1 Gradient BC = 3 2 B1 Uses m1m2 = –1 and equation of perpendicular bisector M1 ( )1 2 3 8 − − = − x y or 3x + 2y = 19 A1 Solve with y = 5 M1 D (3, 5) A1 Complete method for area M1 15 A1 [8] 8 (a) (i)       − 8 10 2 14 6 4 B1 (ii) Matrix multiplication M1       44 64 8 12 A1 (iii) Matrix multiplication M1       − 80 6 22 18 1 5 A1 (b) (i)       − − = − 2 7 1 6 5 1 C 1 or B1+B1 (ii) X = DC–1 M1       − 0 5 2 3 5 1 or       − 0 1 4.0 6.0 A1 [9] www.theallpapers.com

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0606 21 © UCLES 2010 9 (i) 2 π = t B1 ∫ + = c t k t v 2 cos d only M1 t2 cos 2 − A1 Uses both limits M1 4 m A1 (ii) t k t v a 2 cos d d = = only M1 t2 cos 8 A1 7.68 ms–2 A1 [8] 10 (i)       = 24 20 OP B1       = 24 7 PL or       − − = 24 7 LP B1 25 24 7 2 2 = + M1A1       − − = t t PL 6 36 8 23 oe B1 (ii) ( ) ( ) 2 2 2 25 6 36 8 23 = − + − t t M1 Solve 3 term quadratic ( ) [ ] 0 12 8 100 2 = + −t t M1 6 – 2 = 4 hours A1 [8] www.theallpapers.com

Page 7 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0606 21 © UCLES 2010 11E (i) Sector angle = 1.2π B1 12 = OD B1 π 8.0 cos 6 12 2 6 12 2 2 2 × × − + = AD M1 2. 17 = AD A1 Uses ( ) ( ) π π 2.7 2.1 6 = × = s (or 22.6) M1 Complete plan ( ) 6 + + θ r AD or ( ) 6 2.7 2. 17 + + π M1 45.8 A1 (ii) π 8.0 sin 12 6 2 1 × × = ∆AOD M1 21.2 A1 Uses ( ) π 2.1 6 2 1 2 × × = A M1 21.6π or 67.8 or 67.9 A1 89.0 or 89 A1 [12] 11O (i) 8 12 3 d d 2 + − = x x x y M1A1 gradient tangent = –1 A1 tangent y – 8 = –1(x – 1) or x + y = 9 A1 Valid method (e.g. substitute x = 4 in both) M1 Valid conclusion (e.g. y = 5 in both) A1 (ii) x x x x x y 5 4 2 4 d 2 3 4 + + − = ∫ M1A1 Uses limits F(4) – F(1) M1 12.75 A1 Subtracts from area of trapezium (19.5) B1 6.75 A1 [12] www.theallpapers.com